4T Sustainnability Microrandomizad Trial

Power simulations

Published

May 24, 2024

Model

\[ TIR_{ij} = \beta_0 + \gamma_i + \beta_1 Trt_{ij} + \beta_2 Time_{ij} + \beta_3 Trt_{ij} Time_j + e_{ij}, \]

with

\[ \gamma_i \sim N(0, \sigma_\gamma^2), \quad e_{ij} \sim N(0, \sigma_e^2), \quad \gamma_i \perp e_{ij} \]

Also, \(\beta_1 = 0\) since, at baseline, the treatment effect is zero. The correlation between two TIRs for the same patient will be given by

\[ \rho = Cor(TIR_{ij}, TIR_{ij'}) = (1 + (\sigma_e/\sigma_\gamma)^2)^{-1}. \]

In all simulated scenarios, we fixed \(\sigma_e/\sigma_\gamma = 0.5\), such that \(\rho = 0.8\). Also, we assume a baseline TIR before microrandomization of \(\beta_0 = 0.75\) and a baseline time trend of \(\beta_2 = -0.005\) as patients get progressively worse TIR if not intervened.

Thus, using sample sizes of 50, 100, and 200, we simulate

\[ TIR_{ij} = .75 + \gamma_i - 0.005 Time_{ij} + \beta_3 Trt_{ij} Time_j + e_{ij}, \]

under the following scenarios

Scenarios A1-A5

  • \(\beta_1 = 0.005, 0.0025, 0.001, 0.0005, 0.0001\)
  • \(\sigma_e = 0.1\) and \(\sigma_\gamma = 0.2\)

Scenarios B1-B5

  • \(\beta_1 = 0.005, 0.0025, 0.001, 0.0005, 0.0001\)
  • \(\sigma_e = 0.08\) and \(\sigma_\gamma = 0.16\)

Scenarios C1-C2

  • \(\beta_1 = 0\)
  • C1: \(\sigma_e = 0.1\) and \(\sigma_\gamma = 0.2\)
  • C2: \(\sigma_e = 0.08\) and \(\sigma_\gamma = 0.16\)

Figure 1 has data from a non-microrandomized sample of size 100, i.e., patients simply randomized to default or add-on treatment at baseline.

Figure 1: data for a randomized (single randomization at baseline, not microrandomized) sample of 100 patients.

Simulation results

The results below were obtained using \(B = 2000\) replications of the scenarios described above.

Figure 2: average number of participants at risk and add-on treatment provided over the study weeks

Table 1:

(suplementary table) complete simulation results

set sample_size bias empirical_stderr mse cover prop_rej
A1 50 5.809286e-06 -3.566890e-19 8.564424e-08 0.9480 1.0000
A1 100 6.862274e-06 -3.714477e-19 3.827569e-08 0.9470 1.0000
A1 200 1.324313e-06 -1.697725e-19 2.115896e-08 0.9440 1.0000
A2 50 2.625476e-06 -1.228401e-19 6.819272e-08 0.9515 1.0000
A2 100 8.929579e-06 -1.355253e-23 3.301321e-08 0.9495 1.0000
A2 200 -1.113102e-06 -1.064111e-19 1.886067e-08 0.9445 1.0000
A3 50 6.290417e-06 5.586352e-20 6.064241e-08 0.9400 0.9755
A3 100 -7.713217e-07 5.414362e-20 2.775936e-08 0.9530 1.0000
A3 200 -2.213033e-06 -6.380530e-20 1.701592e-08 0.9405 1.0000
A4 50 -2.216276e-06 -4.420834e-20 5.993495e-08 0.9395 0.5540
A4 100 -3.830396e-06 -1.967742e-20 2.940244e-08 0.9430 0.8400
A4 200 -2.526205e-06 2.556007e-20 1.604977e-08 0.9520 0.9730
A5 50 -3.458572e-06 -1.084202e-22 5.652006e-08 0.9485 0.0670
A5 100 4.747583e-06 -7.369187e-21 2.724714e-08 0.9530 0.0870
A5 200 -6.401977e-06 -9.080193e-21 1.626227e-08 0.9470 0.1170
B1 50 1.151939e-05 5.179776e-20 5.203912e-08 0.9470 1.0000
B1 100 -1.410689e-06 -3.453184e-19 2.444409e-08 0.9560 1.0000
B1 200 7.794952e-06 1.048830e-19 1.232330e-08 0.9565 1.0000
B2 50 7.938537e-06 -1.563962e-19 4.698921e-08 0.9455 1.0000
B2 100 4.643525e-06 -2.373048e-19 2.283158e-08 0.9440 1.0000
B2 200 8.703227e-06 -9.451024e-20 1.179911e-08 0.9500 1.0000
B3 50 -1.066334e-06 1.282069e-20 3.607939e-08 0.9505 0.9985
B3 100 4.615945e-06 8.109155e-20 1.790441e-08 0.9485 1.0000
B3 200 3.339763e-06 1.096128e-19 1.015828e-08 0.9495 1.0000
B4 50 8.384059e-07 -5.147250e-20 3.448224e-08 0.9545 0.7635
B4 100 -2.140655e-06 1.124860e-21 1.814290e-08 0.9475 0.9620
B4 200 3.731661e-06 5.139118e-20 1.023060e-08 0.9450 0.9990
B5 50 6.738207e-06 -1.521271e-20 3.614501e-08 0.9400 0.0975
B5 100 -3.289259e-06 -4.160626e-21 1.684663e-08 0.9560 0.1125
B5 200 3.144873e-06 6.993104e-21 1.016923e-08 0.9445 0.1850
C1 50 -3.293292e-06 -3.361027e-21 5.413367e-08 0.9465 0.0515
C1 100 -3.127144e-06 -3.089976e-21 2.898190e-08 0.9460 0.0515
C1 200 -1.940042e-06 1.057097e-21 1.618484e-08 0.9480 0.0510
C2 50 -2.847404e-06 -4.499439e-21 3.517013e-08 0.9480 0.0505
C2 100 -8.824565e-07 8.673617e-22 1.698737e-08 0.9545 0.0440
C2 200 1.054801e-06 5.421011e-23 9.861500e-09 0.9520 0.0475